Using Polarimetry to Detect and Characterize Jupiter-Like Extrasolar Planets

A&A 428, 663–672 (2004) Astronomy DOI: 10.1051/0004-6361:20041578 & c ESO 2004 Astrophysics

Using polarimetry to detect and characterize Jupiter-like extrasolar planets

D. M. Stam1,J.W.Hovenier1, and L. B. F. M. Waters1,2

1 Astronomical Institute “Anton Pannekoek”, University of Amsterdam, Kruislaan 403, 1098 SJ Amsterdam, The Netherlands e-mail: [email protected] 2 Instituut voor Sterrenkunde, K.U. Leuven, Celestijnenlaan 200B, 3001 Heverlee, Belgium

Received 14 April 2004 / Accepted 9 August 2004

Abstract. Using numerical simulations of flux and polarization spectra of visible to near-infrared starlight reflected by Jupiter-like extrasolar planets, we show that polarimetry can be used both for the detection and for the characterization of extrasolar planets. Polarimetry is valuable for detection because direct, unscattered starlight is generally unpolarized, while starlight that has been reflected by a planet will generally be polarized. Polarimetry is valuable for planet characterization because the degree of polarization of starlight that has been reflected by a planet depends strongly on the composition and structure of the planetary atmosphere.

Key words. techniques: polarimetric – stars: planetary systems – polarization

1. Introduction R/D, with R the stellar radius and D the planet’s orbital radius). Clearly, the characterization of an extrasolar planet’s at- Since the discovery of a planet around 51Peg (Mayor & Queloz mosphere requires observing the thermal radiation the planet 1995), the number of known extrasolar planets around solar- emits and/or the stellar light it reflects directly. Besides mak- type stars has increased to more than 120 today. These ex- ing the characterization of atmospheres possible, direct obser- trasolar planets have been discovered using indirect detection vations of planetary radiation will also enable the detection of methods, in which not radiation from the planet itself, but the extrasolar planets that are not easily found with the currently planet’s influence on the star is observed. Given the currently used indirect detection methods, such as planets that are not used detection methods, it is not surprising that the vast major- very massive and/or in wide orbits. ity of the extrasolar planets discovered so far have minimum Direct observations of extrasolar planets are not yet avail- masses that are typical of giant, gaseous planets, and orbital able: they are extremely challenging because the faint signal radii of at most a few astronomical units. Apart from some or- of an extrasolar planet is easily lost in the bright stellar glare. bital parameters and a lower mass limit, little is known about In this paper, we discuss the use of polarimetry both for the the planets’ physical characteristics. A notable exception is the enhancement of the contrast between a planet and its parent planet around HD 209458, that can be observed transiting its star, thus facilitating the detection of an extrasolar planet, and star (see Charbonneau et al. 2000, and references therein). Such for the characterization of the planet. Polarimetry can help a transit not only allows the observer to estimate the plane- planet detection because, integrated over the stellar disk, di- tary size and mass, but it also gives information on upper at- rect starlight can be considered to be unpolarized (Kemp et al. mospheric constituents (Charbonneau et al. 2002; Vidal-Majar 1987), while starlight that has been reflected by the planet et al. 2003, 2004). will generally be polarized. Polarimetry can be used for planet Although transits can provide information on extrasolar characterization because the degree of polarization of the re- planetary atmospheres, as in the case of HD 209458b, they are flected starlight depends strongly on the composition and struc- not suited for the characterization (i.e. the derivation of chemi- ture of the planetary atmosphere. This is well-known from cal compositions, sizes, and spatial distributions of atmospheric remote-sensing of solar system planetary atmospheres, in par- constituents, such as cloud particles) of such atmospheres in ticular Venus (Hansen & Hovenier 1974) and the Earth (see e.g. general; not only because they probe merely the upper atmo- Hansen & Travis 1974, and references therein). spheric layers, but also because the chances of a planet with Because of their sizes and because many of them are al- e.g. a Jupiter-like orbit (about 5.2 AU) being observable in tran- ready known to exist thanks to indirect detection methods, the sit are less than 0.1% (this chance can roughly be described as first extrasolar planets to be detected directly will most likely

Article published by EDP Sciences and available at http://www.aanda.org or http://dx.doi.org/10.1051/0004-6361:20041578 664 D. M. Stam et al.: Polarimetry to detect and characterize Jupiter-like extrasolar planets planet be giant, gaseous planets (Extrasolar Giant Planets or EGPs). In this paper, we use numerical simulations of starlight that is Θ D reflected by Jupiter-like EGPs to illustrate several advantages star of polarimetry for extrasolar planet detection and characteri- α zation. Numerically simulated polarization signals of extraso- d lar giant planets have been presented earlier by Seager et al. (2000), Saar & Seager (2003), and Hough & Lucas (2003). These authors, however, considered so-called close-in EGPs (CEGPs), that have orbital radii ≤0.05 AU, and, consequently, very high temperatures (∼1000 K). Because of the small orbital Earth radii, it will not be possible to spatially resolve the (polarized) Fig. 1. Distances and angles involved in observing an extrasolar light reflected by such a close-in EGP from the (unpolarized) planet: α is the phase angle, Θ (=π − α) is the scattering angle direct light of its parent star. As a result, the degree of polar- (0 ≤ Θ ≤ π), D is the distance between the star and the planet, and d is ization that one can measure for such an unresolved planetary the distance between the planet and the Earth. Note that in our numer- system is expected to be very small (∼10−5) (see Seager et al. ical simulations of starlight reflected by the planet, we assume d r 2000; Saar & Seager 2003; Hough & Lucas 2003). and D R, with r the radius of the planet, and R the stellar radius. Because P is a relative measure, it can be determined very accurately: the accuracy of sophisticated polarimeters is − of the order of 10 5 (Povel 1995), and more sensitive (better in the difference between the images. CHEOPS’ polarimetry is − than 10 6) polarimeters are being built with the detection of ex- planned to be performed in the near-infrared, across the I-band. trasolar planets in mind (see Hough & Lucas 2003). Although In Sect. 2, we introduce the Stokes vector description of such sensitivities are imperative for planet detection in unre- planetary radiation and polarization. In Sect. 3, we describe solved planetary systems, the mere detection of an EGP in a the radiative transfer algorithm and the atmospheric models Jupiter-like orbit that can be spatially resolved from its star can of the Jupiter-like EGPs we use in the numerical simulations. be performed with a less accurate polarimeter. However, when Section 4 contains the numerically simulated total and polar- observing such EGPs, the use of an accurate polarimeter is ized fluxes of the EGPs, and in Sect. 5 we discuss and summa- strongly recommended, because, as we will illustrate in this pa- rize the advantages of polarimetry for extrasolar planet detec- per, polarimetry is a valuable tool not only for detecting EGPs, tion and characterization. but also for the characterization of their atmospheres (i.e. the derivation of chemical compositions, sizes, and spatial distributions of atmospheric constituents, such as cloud particles), 2. Planetary radiation In this paper, we concentrate on polarization signals of 2.1. Stokes vectors and polarization EGPs at Jupiter-like distances (several astronomical units) from their star. In the near-future, such planets may be spatially re- The flux (irradiance) and state of polarization of planetary radi- λ solved from their star, for example, from space, or from the ation with wavelength can be described by a Stokes (column) ground in combination with adaptive optics systems to correct vector F (see Hovenier & van der Mee 1983) for atmospheric turbulence. With a spatially resolved planetary F(λ, α) = [F(λ, α), Q(λ, α), U(λ, α), V(λ, α)], (1) system, in which the polarized planetary light can be observed without dilution by the unpolarized stellar light, it will be pos- with α the planetary phase angle, i.e. the angle between the sible to measure high degrees of polarization, as we will show star and the observer as seen from the center of the planet here. We will discuss the dependence of the polarization of the (see Fig. 1). Here, the Stokes parameter F describes the to- planetary light on the planetary atmosphere, on the geometry, tal, Q and U the linearly polarized, and V the circularly po- and, in particular, on the wavelength. This spectral dependence larized flux. All Stokes parameters in Eq. (1) have the dimen- is important for determining optimal instrumental wavelength sion W m−2 m−1. / regions for planet detection and or planet characterization. Stokes parameters Q and U are defined with respect to a An example of an instrument for direct detection of EGPs reference plane. We chose our reference plane through the cen- that combines polarimetry, spectroscopy, and adaptive op- ters of the star, the planet, and the observer. This plane is re- tics, is CHEOPS (CHaracterizing Exoplanets by Opto-infrared ferred to as the planetary scattering plane. Stokes vectors can Polarimetry and Spectroscopy) (Feldt et al. 2003) that has be transformed from this reference plane to another, e.g. the op- been proposed to ESO for use on the VLT. CHEOPS will use tical plane of a polarimeter, using the following rotation matrix the well-established and highly accurate (10−5) Zürich imag- (see Hovenier & van der Mee 1983) ing polarimeter (ZIMPOL) (Povel et al. 1990; Povel 1995)    10 00 for detection and characterization of extrasolar planets. With   /   CHEOPS ZIMPOL a (suspected) planetary system will be im-  0cos2β sin 2β 0  β =   . aged in two perpendicular polarization directions. Unpolarized L( )   (2)  0 − sin 2β cos 2β 0  sources, like the star, will be equally bright in these two im-   ages, whereas a polarized source, like a planet, will show up 00 01 D. M. Stam et al.: Polarimetry to detect and characterize Jupiter-like extrasolar planets 665

Angle β is the angle between the two reference planes, mea- this reference plane, the planetary scattering matrix S is given sured rotating in the anti-clockwise direction from the original by (see Hovenier 1969) to the new reference plane when looking towards the observer.    λ, α λ, α  ThedegreeofpolarizationP of the radiation described by  a1( ) b1( )0 0    λ, α λ, α  Stokes vector F (see Eq. (1)), is defined as  b1( ) a2( )0 0 S(λ, α) =   . (6)  λ, α λ, α   00a3( ) b2( )  Q2(λ, α) + U2(λ, α) + V2(λ, α)   P(λ, α) = · (3) − λ, α λ, α F(λ, α) 00b2( ) a4( ) With unpolarized incident starlight, the only elements of S that Assuming that the planet is mirror-symmetric with respect to are needed to calculate the Stokes vector F of the light that the reference plane, Stokes parameters U and V are zero when is reflected by the planet are a and b (cf. Eq. (5)). Starlight integrated over the planetary disk. In that case, the degree of 1 1 that has been diffracted in the planetary atmosphere, and that polarization of the reflected light can simply be defined as ◦ might be observed when α = 180 (in the case of a transiting extrasolar planet), is not included in S. λ, α = − λ, α / λ, α . P( ) Q( ) F( ) (4) Matrix S is normalized so that the average of the planetary phase function, which is represented by matrix element a1, In Eq. (4), we have included the minus sign to add information over all directions equals the planet’s Bond (or spherical) about the direction of polarization: for P > 0(P < 0), the light albedo AB,i.e. is polarized perpendicular (parallel) to the reference plane. 1 1 π a1(λ, α)dω = a1(λ, α)sinα dα ≡ AB(λ), (7) 4π π 2 2.2. Reflected starlight 4 0 where dω is an element of solid angle. The Bond albedo is Given a spherical extrasolar planet with radius r, the planetary the fraction of the irradiance of the incident starlight that is λ radiation (see Eq. (1)) with wavelength that has been reflected reflected by the planet in all directions. The planetary Bond by the planet and that arrives at a distance d (with d r) can albedo plays an important role in determining a planet’s energy be described by balance, because 1 − AB indicates how much stellar light is being absorbed by the planet. 2 2 r R 1 For observers, the geometric albedo of a planet is more in- F(λ, α) = S(λ, α) πB0(λ). (5) d2 D2 4 teresting than the Bond albedo, because it is a measure of the brightness of a planet. The geometric albedo AG of a planet In Eq. (5), R is the stellar radius, and D the distance between the is defined as the ratio of the flux that is reflected by the star and the planet (with D R). Figure 1 shows the phase an- planet at opposition (when α = 0◦) to the flux reflected by a gle and the distances appearing in Eq. (5). Note that, analogous Lambertian surface (i.e. a surface that reflects all incoming ra- ◦ − α =Θ Θ to the scattering of light by a particle, 180 , with diation isotropically and completely depolarized) that receives the so-called total scattering angle. Furthermore in Eq. (5), S is the same incoming irradiance and that subtends the same solid the planetary scattering matrix, and B0 represents the Stokes angle (i.e. πr2/d2) on the sky. Thus, (column) vector [B0, 0, 0, 0] = B0 [1, 0, 0, 0], with πB0 the stel- − − lar surface flux (in W m 2 m 1). Integrated over the stellar F(λ, 0◦) D2 d2 1 A λ = = a λ, ◦ . G( ) 2 2 1( 0 ) (8) disk, this flux can be considered to be unpolarized (Kemp et al. B0(λ) R πr 4 1987), hence the use of the unit column vector. The starlight is assumed to be unidirectional (D R) when it arrives at Note that the ratio of the Bond albedo to the geometric albedo the planet. is the so-called phase integral. Of course, to derive the absolute The planetary scattering matrix S describes the light that brightness of a planet, knowing AG is not enough; proper values has been scattered within the planetary atmosphere and that is of r, d, R, D,andB0 (see Eq. (5)) should also be available. reflected towards the observer. Because for years to come, extrasolar planets will be observable as point sources only (once 2.3. Thermally emitted radiation they are directly observable), S describes the reflected starlight integrated over the illuminated and visible part of the plane- Equation 5 describes the starlight that is reflected by the planet. tary disk. Matrix S will generally depend strongly on the wave- Planetary radiation, however, not only consists of reflected length λ of the reflected light, through the composition and starlight, but also of thermal radiation that is emitted by the structure of the planetary atmosphere and the optical proper- planet. This thermal radiation is generally emitted at (near-) ties of the atmospheric constituents like gases, aerosol, and infrared wavelengths, and it will be unpolarized, in particular cloud particles (see Sect. 3), and on the planetary phase an- when integrated over the planetary disk. The degree of polar- gle α, which determines which fraction of the illuminated side ization of the reflected plus the thermal planetary radiation thus of the planet can actually be observed (see Sect. 4). equals (cf. Eq. (4)) Using the planetary scattering plane as the reference plane, λ, α λ, α = − Qreflected( ) , and assuming a planet that is mirror-symmetric with respect to P( ) (9) Freflected(λ, α) + Fthermal(λ, α) 666 D. M. Stam et al.: Polarimetry to detect and characterize Jupiter-like extrasolar planets

-2 10 In Fig. 2, the two contributions to the planetary radiation -4 can easily be distinguished: at the shortest wavelengths, the 10

) star planetary spectrum is dominated by reﬂected stellar radiation,

-1 -6 10 and at the longest wavelengths, by the planet’s thermal ra- m

-2 -8 diation. For very hot planets or, for example, brown dwarfs, 10 the thermally emitted radiation will limit the use of polarime- -10 10 planet try to visible wavelengths (up to about 0.8 µm for a planet -12 at 1000 K). As can be derived from the spacing of the lines 10 Flux (W m in Fig. 2, Jupiter-like planets cool down relatively rapidly. The -14 10 wavelength region available for polarimetry thus increases rel- -16 atively rapidly towards the infrared. For a 4 × 109 years old 10 0.1 1.0 10.0 100.0 Wavelength (µm) Jupiter-like planet, reflected radiation will dominate the planetary radiation, and hence the degree of polarization of the plan- Fig. 2. The flux of a solar-type star (upper solid line) (with an effec- etary radiation, for wavelengths up to about 6.0 µm. tive temperature of 6000 K, and a radius of 700 000 km) at 5 pc, and In Fig. 2, the planetary geometric albedo AG was set equal the planetary flux (lower lines) of a Jupiter-like planet (with a ra- to 1. With this (wavelength independent) value of AG,the dius of 70 000 km, and a geometric albedo of 1.0) at 5.2 AU from (wavelength independent) ratio of the reflected stellar flux to its star, with an age of 0.125 (solid line), 0.25, 0.5, 1.0, 2.0, 4.0, the direct stellar flux in Fig. 2 is 0.8 × 10−8. In reality, this and 8.0 (dot-dash-dash line) Gyrs. The corresponding effective tem- ratio, the albedo and, more generally, the planetary scattering peratures of the planet are respectively 268.8 (solid line), 225.6, 189.3, 159.4, 133.2, 110.4, and 91.2 K (dot-dash-dash line) (the age – tem- matrix S, depend on the composition and structure of the plan- λ perature relations have been taken from Burrows et al. 2003). The star etary atmosphere, the wavelength , and the planetary phase α and planet are assumed to radiate like black bodies, and spectral fea- angle (AG is phase angle independent since it has been de- ◦ tures, e.g. due to absorbing gases and/or absorbing and scattering at- fined for α = 0 ). In this paper, we will calculate S, and from mospheric particles, are ignored. that, F and P, for Jupiter-like extrasolar planets from visible to near-infrared wavelengths, in particular from 0.4 to 1.0 µm. According to Fig. 2, we can safely ignore thermally emitted planetary radiation across this wavelength region. with Freflected and Qreflected given by Eq. (5). The contribution of the thermally emitted flux to the total planetary flux, and thus 3. Radiative transfer calculations to the decrease of the degree of polarization P of the planetary radiation, depends strongly on the temperature of the planet. 3.1. Calculating the planetary scattering matrix S Figure 2 shows, schematically, the flux of a solar-type star The first step towards calculating S is the modeling of plane- and the total planetary flux (F + F ) of a Jupiter-like reflected thermal tary atmospheres as stacks of locally plane-parallel, horizon- planet at various ages (see Burrows et al. 2003). The planet’s tally homogeneous layers, containing gaseous molecules, and, temperature is assumed to be solely determined by the planet’s optionally, aerosol particles. A model atmosphere is bounded age; any temperature increase due to absorption of stellar radi- below by a black surface (i.e. all the light that escapes the at- ation is neglected. This is a reasonable assumption for a planet mosphere at the bottom is assumed to be absorbed). in a not-to-close orbit around its star, like the Jupiter-like plan- Directions in the plane-parallel model atmosphere are spec- ets we consider in this paper (remember that planets that do ified by the cosine of the angle with the downward vertical, heat up significantly because of the absorption of stellar radi- µ = cos θ, and the azimuthal angle, φ. The azimuthal an- ation, the so-called “roasters”, will be too close to their star to gle is measured from an arbitrary plane and clockwise when be spatially resolvable). Furthermore, in Fig. 2, the star and the looking upward. The direction of the incident starlight is de- planet are assumed to radiate like perfect black-bodies; spec- noted by (µ ,φ ), and that of the light reflected towards the ob- tral features due to e.g. absorbing gases and/or scattering and 0 0 server by (µ, φ). Since we assume horizontally homogeneous absorbing atmospheric particles have been omitted. The phase ◦ planetary atmospheres, only the azimuthal difference φ − φ is angle α of the planet is 0 , and the planet’s (wavelength in- 0 relevant. dependent) geometric albedo AG is 1.0. Assuming unpolarized incoming stellar light, the flux that is reflected by the planet Given a plane-parallel model atmosphere, the second step (in Fig. 2) is then given by (cf. Eqs. (5) and (8)) towards calculating S, is the computation of the local reflection matrix R for various combinations of (µ0,φ0)and(µ, φ). 2 2 2 2 For this, we use an efficient adding-doubling radiative transfer ◦ r R 1 ◦ r R F(λ, 0 ) = a1(λ, 0 ) πB0(λ) = πB0(λ). (10) algorithm (van de Hulst 1980) that fully includes multiple scat- d2 D2 4 d2 D2 tering and polarization (de Haan et al. 1987). For each plan- The solar-type star has an effective temperature of 6000 K, etary phase angle α, the various local reflection matrices are and a radius R of 700 000 km. The planet has a radius r integrated over the illuminated and visible part of the planetary of 70 000 km, and orbits its star at a distance D of 5.2 AU. disk to obtain the planetary reflection matrix S. Details of the The planetary system is located at a distance d of 5 pc from the computation of the planetary scattering matrix will be given in observer. Stam et al. (in preparation). D. M. Stam et al.: Polarimetry to detect and characterize Jupiter-like extrasolar planets 667

2 10 1.0 a. molecules b. cloud particles 0.5 1 10 haze particles 0.0

0 10 -0.5 Normalized phase function -1 Degree of linear polarization -1.0 10 03060 90 120 150 180 03060 90 120 150 180 Scattering angle Θ (degrees) Scattering angle Θ (degrees)

Fig. 3. The phase functions and the degree of linear polarization of light singly scattered by gaseous molecules (solid lines), the cloud particles (dashed lines), and the haze particles (dotted lines) at a wavelength λ = 0.7 µm as functions of the scattering angle Θ (in degrees) (when Θ=180◦, the light is scattered in the backward direction). The phase functions are normalized such that the average over all directions equals unity.

3.2. The model atmospheres scattering matrix of the cloud and haze particles are calculated using Mie-theory (van de Hulst 1957; de Rooij & van der Stap The adding-doubling radiative transfer calculations for the lo- 1984). Figure 3a shows the phase function and Fig. 3b the de- cal reflection matrices R require us to specify for each atmo- gree of linear polarization P (see Eq. (4)) for incoming unpolar- spheric layer and for each wavelength: the optical thickness, ized light with λ = 0.7 µm that has been singly scattered by, re- the single scattering albedo, and the scattering matrix of the spectively, the gaseous molecules, the cloud, and the haze par- mixture of gaseous molecules and aerosol particles in the layer ticles. Here, P > 0(P < 0) indicates scattered light that is po- (see Stam et al. 1999). larized perpendicular (parallel) to the scattering plane (i.e. the The molecular optical thickness of an atmospheric layer de- plane containing the incoming as well as the scattered light). pends on the ambient pressure and temperature. For this paper, The wiggles in the curves pertaining to the haze particles are we use model atmospheres consisting of 38 layers and hav- due to the narrowness of the size distribution (see Hansen & ing a Jupiter-like atmospheric pressure and temperature pro- Travis 1974). Figure 3b clearly shows the strong polarizing ca- file. The ambient pressure ranges from 1.0 × 10−4 bars at the pabilities of gaseous molecules, in particular for scattering an- top to 5.623 bars at the bottom of our model atmospheres gles Θ around 90◦. (Lindal 1992, supplemented from 1.0 to 5.623 bars by data from West et al. 1986). Across the wavelength region of our Note that the scattering matrix elements of the cloud and haze particles will be wavelength dependent, whereas those of interest (from 0.4 to 1.0 µm), methane (CH4)isthemainab- sorbing gas in a Jupiter-like atmosphere. We assume an atmo- the gaseous molecules are not (ignoring the slight wavelength dependence of the depolarization factor). spheric mixing ratio of CH4 of 0.18%, like on Jupiter, and adopt the absorption cross-sections of Karkoschka (1994). The molecular scattering optical thickness of the model at- In this paper, we will present results of numerical simula- mospheres decreases from 21.47 at 0.4 µmto0.51at1.0µm. tions for three types of planetary model atmospheres: model 1 The tropospheric cloud (models 2 and 3) is located in the at- contains only gaseous molecules (the “clear atmosphere”), mospheric layers between 5.623 and 1.0 bars. Its (scattering) model 2 is identical to model 1 except for a tropospheric cloud optical thickness is chosen to be equal to 6.0 at 0.7 µm(the layer (the “low cloud atmosphere”), and model 3 is identical to cloud optical thickness of each layer is chosen proportional to model 2 except for a stratospheric haze layer (the “high haze the molecular scattering optical thickness of the layer). With atmosphere”). The cloud and haze particles are homogeneous, the size distribution of the cloud particles as described above, spherical particles distributed in size according to the standard the cloud’s (scattering) optical thickness increases smoothly distribution of Hansen & Travis (1974). For the cloud particles, from 5.45 at 0.4 µm to 7.34 at 1.0 µm. The stratospheric haze the effective radius, reff, is taken equal to 1.0 µmandtheeffec- (model 3) is located in the atmospheric layers between 0.133 tive variance, veff, is 0.1. For the haze particles, reff = 0.5 µm and 0.1 bars, with a (scattering) optical thickness of 0.25 µ and veff = 0.01 (Sromovsky & Fry 2002). The refractive in- at 0.7 m (Sromovsky & Fry 2002). The haze optical thickness µ µ dex of the cloud particles is 1.42, which is typical for NH3 ice varies from 0.23 at 0.4 m to 0.35 at 1.0 m. at 0.7 µm (Martonchik et al. 1984). For the refractive index of As we know from observations of Jupiter in our own Solar the haze particles we use 1.66 (following Sromovsky & Fry System, the atmosphere of a Jupiter-like EGP will undoubtedly 2002). be much more complex than the model atmospheres considered The scattering of the gaseous molecules is described by in this paper: there might be latitudinal bands consisting of var- anisotropic Rayleigh scattering, using the depolarization factor ious cloud layers composed of numerous types of cloud parti- of H2, the major constituent of a Jupiter-like atmosphere, cles (see e.g. Sromovsky & Fry (2002) and references therein). i.e. 0.02 (see Hansen & Travis 1974). The wavelength de- Our simple model atmospheres, however, suffice to illustrate pendent extinction cross-section, single scattering albedo, and both the advantages of polarimetry for the detection of EGPs 668 D. M. Stam et al.: Polarimetry to detect and characterize Jupiter-like extrasolar planets

0.20 1.0

a. P b. 0.8 0.15

F 0.6 0.10

Flux model 1 0.4 0.05 model 2 model 3 0.2 Degree of polarization 0.00 0.0 0.4 0.5 0.6 0.7 0.8 0.9 1.0 0.4 0.5 0.6 0.7 0.8 0.9 1.0 Wavelength (µm) Wavelength (µm)

Fig. 4. The ﬂux F and degree of polarization P of starlight reﬂected by three Jupiter-like EGPs for α = 90◦. Planetary model atmosphere 1 (solid lines) contains only molecules, model 2 (dashed lines) is similar to model 1, except for a tropospheric cloud layer, and model 3 (dotted lines) is similar to model 2, except for a stratospheric haze layer.

and the sensitivity of the degree of polarization of the reflected For model 2, which is the atmosphere with the tropospheric stellar radiation to the structure and composition of the plane- cloud, both F and P (Fig. 4a and 4b) at the shortest wave- tary model atmosphere. lengths are similar to those of model 1, because at these wavelengths, the molecular scattering optical thickness of the atmospheric layers above the cloud is so large that hardly any stellar 4. Results light will reach the cloud layer. In the strong CH4-absorption band around 0.89 µm, With increasing wavelength, the molec- 4.1. Flux and polarization as functions of wavelength ular scattering optical thickness decreases, and, at least at con- Figure 4 shows spectra of the reflected flux F and degree of po- tinuum wavelengths, the contribution of light scattered by the larization P for the 3 model atmospheres and a planetary phase cloud particles to the reflected F and P increases. The slope of angle α of 90◦. These spectra have been calculated at the same the continuum F is less steep for model 2 than for model 1, because while the molecular scattering optical thickness de- 1-nm intervals at which the CH4 absorption cross-sections have been given (Karkoschka 1994). In order to present general re- creases with wavelength, the cloud’s (scattering) optical thick- 2 2 2 2 ness increases. The decrease of the continuum P for model 2 sults in Fig. 4, we have set πB0r R /(d D ) (see Eq. (5)) equal 1 λ, ◦ (Fig. 4b) is due to the increased multiple scattering within the to one. In fact, Fig. 4a thus shows 4 a1( 90 ), with a1 the (1,1)-element of the planetary scattering matrix S (cf. Eq. (6)) cloud layers, as well as to the low degree of polarization of light ◦ ◦ that is scattered by cloud particles (see Fig. 3b). and Fig. 4b, −b1(λ, 90 )/a1(λ, 90 ). Using Eq. (5), scaling the fluxes presented in Fig. 4a to obtain results for a Jupiter-like In the CH4-absorption bands, F is generally larger and P extrasolar planet in an arbitrary planetary system is a straight- smaller for model 2 than for model 1, just like at continuum forward excercise. Note that P (Fig. 4b) is independent of the wavelengths. In the strong absorption band around 0.89 µm, choice of r, R, d, D,andB0, because P is a relative measure. however, F and P of the two models are similar, because at The flux and polarization spectra in Fig. 4 can be thought these wavelengths, hardly any stellar light can reach the cloud of to consist of a continuum with superimposed high-spectral layer due to the large molecular absorption optical thickness resolution features that are due to absorption by CH4. Recent of the atmosphere above the cloud. The light that is reflected Earth-based spectropolarimetric measurements of the gaseous at these wavelengths, has thus been scattered in the highest at- planets of our own Solar System using ZIMPOL show a similar mospheric layers, which are identical in model atmospheres 1 spectral structure (Joos et al. 2004). and 2. For model 1, which is the clear atmosphere, the continuum For model 3, the atmosphere with the tropospheric cloud F decreases steadily with λ (Fig. 4a), following the decrease and the stratospheric haze, F is at all wavelengths somewhat of the molecular scattering optical thickness with λ. The con- larger than for model 2, even at the shortest wavelengths, where tinuum P (Fig. 4b) increases with λ, because the smaller the model 2 is almost indistinguishable from model 1 (Fig. 4a). molecular optical thickness, the less multiple scattering takes The influence of the optically thin haze on F is explained by place within the atmosphere; and multiple scattering tends to the relatively small molecular scattering and absorption optical lower the degree of polarization of the reflected light. Multiple thickness above the high-altitude haze layer: at all wavelengths, scattering also decreases with increasing absorption by CH4. a significant fraction of the incoming stellar light reaches the For model atmosphere 1, this fully explains the high values haze layer and is reflected back to space. The degree of polar- of P within the CH4 absorption bands (Stam et al. 1999). In ization P (Fig. 4b) for model 3 is at all wavelengths signifi- the strong absorption band around 0.89 µm, P = 0.95, and thus cantly lower than that for model 1 and 2 mainly because light almost reaches its single scattering value at a single scattering that is singly scattered by the haze particles has a very low de- angle of 90◦ (which corresponds with a planetary phase angle α gree of polarization (see Fig. 3b). In particular, P is very low of 90◦), namely 0.96 (see Fig. 3b). in the strong absorption band around 0.89 µm. Whereas in the D. M. Stam et al.: Polarimetry to detect and characterize Jupiter-like extrasolar planets 669

0.5 0.6

a. model 1 P 0.5 b. 0.4 model 2 model 3 0.4

F 0.3 0.3 0.2 Flux 0.2 0.1 0.1 0.0 Degree of polarization 0.0 -0.1 03060 90 120 150 180 03060 90 120 150 180 Phase angle (degrees) Phase angle (degrees)

Fig. 5. The flux F and degree of polarization P of the reflected starlight for the three model atmospheres as functions of the planetary phase angle α, averaged over the wavelength region between 0.65 and 0.95 µm. clear and cloudy atmospheres of models 1 and 2, the reflected 1.0 light in this band has been scattered by molecules, in the atmo- model 1 sphere of model 3, the depolarizing haze particles are the main F 0.8 model 2 scatterers. model 3 0.6 4.2. Flux and polarization as functions of phase angle 0.4 The flux and degree of polarization of reflected starlight not only depend on the planetary atmosphere, but also on the plane- Normalized flux 0.2 tary phase angle α. As seen from the Earth, the gaseous planets 0.0 of our own Solar System can only be observed at small phase 03060 90 120 150 180 angles (the maximum phase angle for Jupiter is, for example, Phase angle (degrees) about 11◦, and for Saturn, 6◦). The phase angles at which an extrasolar planet can be observed along its orbit range from 90◦−i Fig. 6. The reflected flux F for the three model atmospheres as ◦ ◦ α to 90 +i, with i the orbital inclination angle (i = 0 for an orbit functions of , averaged over the wavelength region between 0.65 µ α = ◦ that is viewed face-on). When α ≈ 0◦ or α ≈ 180◦, it will be im- and 0.95 m, and normalized at 0 . possible to spatially separate the extrasolar planet from its star / and thus to measure F and or P of the reflected starlight with- This is illustrated in Fig. 6, where we have plotted the fluxes for out including the direct stellar light, because in those cases the the three model atmospheres normalized at α = 0◦, thus in fact, planet is located either behind or in front of the star. For com- (see Eq. (8)) divided by the geometric albedo AG For model pleteness, we do include these phase angles in our calculations. atmosphere 1, we find AG = 0.31, for model 2, AG = 0.39, π 2 2/ 2 2 Figure 5 shows F (assuming B0r R (d D ) is equal to 1) and for model 3, AG = 0.43. From the curves in Fig. 6, we can α 2 2 2 2 and P for the three model atmospheres as functions of ,aver- conclude that without accurate information on πB0r R /(d D ) aged over the wavelength region between 0.65 and 0.95 µm. of an extrasolar planetary system, measuring F will yield little We average over a wavelength region rather than present information about the atmosphere, even when the planet is ob- monochromatic results because due to the faintness of extra- served at different phase angles. solar planets, the first direct observations will probably be per- The phase angle dependence of P depends strongly on formed using broadband filters. We average over this particu- the model atmosphere, except for reflection near the backward lar wavelength region (the I-band) because that is where po- (α = 0◦) and forward (α = 180◦) directions, where P is zero larimetry with the CHEOPS instrument (Feldt et al. 2003) and regardless of the model atmosphere, because of symmetry prin- ZIMPOL (Povel et al. 1990; Povel 1995) for the VLT (Feldt ciples. As explained before, such small and large phase angles et al. 2003) is planned to take place. will not be accessible for polarimetry of spatially resolved ex- 1 α = ◦ The reflected flux in Fig. 5 equals 4 a1, and thus, at 0 trasolar planets anyway, because the planet will be either be- the planet’s geometric albedo AG (in this case averaged over hind or in front of the star. From the Earth, the gaseous planets the wavelength region between 0.65 and 0.95 µm). As can be of our own Solar System are only observable at small phase seen in Fig. 5a, the reflected fluxes F are smooth functions angles. Earth-based polarimetry of these planets therefore al- of α. Only the curve for model atmosphere 2, with the tropo- ways yields low values of P (see Joos et al. 2004 for recent spheric cloud layer, shows a slight depression at small phase Earth-based polarimetric measurements of Jupiter, Uranus and angles. This depression can be attributed to the local mini- Neptune). mum in the single scattering phase function of the cloud parti- For all three model atmospheres in Fig. 5b, P peaks near cles (see Fig. 3a) for scattering angles between 165◦ and 180◦. α = 90◦, because P of the light that has been singly scattered Apart from the albedo, the reflected flux appears to be insensi- by the gaseous molecules, which are present in each of the tive to the composition and structure of the model atmosphere. model atmospheres, is largest at a scattering angle of 90◦ (see 670 D. M. Stam et al.: Polarimetry to detect and characterize Jupiter-like extrasolar planets

0.5 0.6

a. P 0.5 b. 0.4 o 0 0.4 o F 0.3 30 0.3 o 60 0.2 Flux 0.2 o 90 0.1 0.1 0.0 Degree of polarization 0.0 -0.1 0.0 0.2 0.4 0.6 0.8 1.0 0.0 0.2 0.4 0.6 0.8 1.0 Orbital period Orbital period

Fig. 7. The flux F and degree of polarization P of the reflected starlight averaged over the wavelength region between 0.65 and 0.95 µm, for model atmosphere 1 and different orbital inclination angles: 0◦ (dot-dashed line), 30◦ (dashed line), 60◦ (dotted line), and 90◦ (solid line).

0.5 0.6

a. P 0.5 b. 0.4 model 1 0.4 model 2

F 0.3 model 3 0.3 0.2 Flux 0.2 0.1 0.1 0.0 Degree of polarization 0.0 -0.1 0.0 0.2 0.4 0.6 0.8 1.0 0.0 0.2 0.4 0.6 0.8 1.0 Orbital period Orbital period

Fig. 8. The ﬂux F and degree of polarization P of the reﬂected starlight averaged over the wavelength region between 0.65 and 0.95 µm, for an inclination angle of 90◦ and the three model atmospheres.

Fig. 3b). For models 1 and 2, P > 0 for phase angles smaller at 0.0 and 0.5 orbital periods, when α = 90◦. The polarization than about 165◦, and the reflected starlight is thus polarized per- curve during the first half of the orbit (from 0.0 to 0.5) differs pendicular to the planetary scattering plane. For larger phase slightly from that during the second half (from 0.5 to 1.0), re- angles (when only a narrow crescent of the planet is visible), flecting the asymmetry of the model 1 curve in Fig. 5b. When P < 0 and the direction of polarization is thus parallel to the i = 0◦, the observed fraction of the planet that is illuminated is scattering plane. This negative polarization is characteristic for constant, and F is thus constant. P’s absolute value is also con- light that has been scattered twice by molecules in the plane- stant, but the direction of polarization, that will generally be tary atmosphere. For these large phase angles, the contribution perpendicular or parallel to the planetary scattering plane, can of the twice scattered light to P can be significant because the be seen to rotate with the planet as it orbits the star. The vari- single scattered light is virtually unpolarized. For model 3, with ation of P along the planetary orbit, either in absolute value or the high altitude haze, P < 0 across a broader range of phase in direction, would help to distinguish the signal of the planet angles, because the haze particles themselves scatter negatively from possible background polarization signals, like that from polarized light (see Fig. 3b). zodiacal dust. Interestingly, unlike the reflected stellar flux (Fig. 7a), the maximum degree of polarization that can be measured along 4.3. Flux and polarization as functions of orbital period the planetary orbit is independent of the orbital inclination angle (Fig. 7b). Because this maximum value does depend on α As said before (in Sect. 4.2), the phase angles an extra- the planetary atmosphere (see Fig. 5b), polarimetry could thus solar planet can be observed at when it orbits its star range provide information about the planetary atmosphere without ◦ − ◦ + from 90 i to 90 i with i the orbital inclination angle. knowledge on the inclination angle. In Fig. 7, we have plotted F and P for the clear atmosphere The value of polarimetry along a planetary orbit for the (model 1) during an orbital period for various inclination angles characterization of EGPs is further illustrated in Fig. 8, which π 2 2/ 2 2 and a circular orbit. Like before, B0r R (d D ) is assumed to shows F and P along the planetary orbit for an inclination an- equal 1. gle of 90◦ and the three model atmospheres. Because the in- Figure 7 shows that except when i ≈ 0◦ (when the orbit clination angle is 90◦, all three polarization curves in Fig. 8b is seen face-on), F and P vary significantly during an orbital are zero at 0.25 and 0.75 orbital periods, when, respectively, period: F is maximum when the planet’s day-side is observed the planet’s night (α = 180◦) and dayside (α = 0◦) are turned (at 0.75 orbital periods in Fig. 7) and zero when the planet’s towards the observer. Clearly, the maximum P of each curve night-side is in view (at 0.25 orbital periods), while P peaks depends strongly on the planetary atmosphere. Obviously, the D. M. Stam et al.: Polarimetry to detect and characterize Jupiter-like extrasolar planets 671 maximum value of F along the planetary orbit also depends This application of polarimetry is illustrated by the sensi- on the planetary atmosphere (Fig. 8a). However, the maximum tivity of P to the three model atmospheres in Figs. 4 and 5: value of F will provide far less information on the planetary at- both the degree of polarization in the continuum and across the mosphere than that of P, because 1) F’s maximum depends on gaseous absorption bands depend strongly on the atmospheric the orbital inclination angle; 2) F is less sensitive to the com- composition and structure. Because extrasolar planets are ex- position and structure of the planetary atmosphere; 3) F can be tremely faint, the spectral resolution of the first direct obser- measured less accurately than P because P is a relative mea- vations of extrasolar planets will probably be low, and would sure; and 4) to derive information on the planet from measured yield the continuum polarization. At short wavelengths, where values of F, accurate knowledge about the planetary system the atmospheric molecular scattering optical thickness is large, 2 2 2 2 (i.e. πB0r R /(d D )) is required. the continuum polarization generally contains information on the upper atmospheric layers. At longer wavelengths, the atmospheric molecular scattering optical thickness is small, and 5. Discussion and summary the continuum polarization depends mainly on the scatterers In this paper, we present numerical simulations of the flux in the deeper atmospheric layers, e.g. cloud particles. Across a and degree of polarization of starlight that is reflected by gaseous absorption band, P contains information on particles Jupiter-like EGPs, to illustrate the advantages of polarimetry at various altitudes in the atmosphere, and thus on the vertical for the detection and characterization of such planets. The di- structure of the atmosphere (Stam et al. 1999). Although the in- rect, unscattered (and unpolarized) starlight is not included in formation content of P across gaseous absorption bands is very our simulations. Our simulations are thus representative of di- high, the spectral resolution that is required for such observa- rect observations of extrasolar planets that orbit their star at tions makes them extremely challenging for years to come. Jupiter-like distances, with fluxes that can be observed spatially Compared to flux observations, another advantage of po- resolved from the stellar fluxes. Such observations may be car- larimetry for the characterization of planetary atmospheres is ried out with e.g. adaptive optics systems on ground-based tele- that because P is a relative measure, it is independent of the scopes, such as CHEOPS (Feldt et al. 2003) that is planned for distances between the planet and the star (D) and between the use on the VLT. planet and the observer (d), the sizes of the planet (r)and The main advantage of using polarimetry for the detec- the star (R), and the incoming stellar flux (B0). So, when these tion of extrasolar planets is that it can be used to distinguish parameters are not accurately known, as will often be the case the weak, polarized signal of starlight that is reflected by a with extrasolar planets, one can still derive atmospheric infor- planet from the strong, unpolarized, direct starlight. As such, mation from P (see Fig. 5b). Obtaining atmospheric informa- polarimetry can be used for direct detections of planets that tion from reflected flux measurements appears to be a much cannot be observed spatially resolved from their star (either be- harder task, because it requires an accurate measurement of the cause they are in a very close orbit, and/or because the star is absolute flux, as can be concluded from Figs. 5a and 6. too far away) (see Seager et al. 2000; Saar & Seager 2003; Finally, because P is a relative measure, it is unaffected Hough & Lucas 2003), and for planets that can be spatially upon transmission through the Earth’s atmosphere. The wave- resolved. We concentrate on spatially resolvable Jupiter-like length dependence of P, and in particular that across gaseous planets. Apart from their orbital distance, these planets differ absorption bands, is thus retained without the need for atmo- from close-in giant planets in their atmospheric composition spheric corrections. This advantage of polarimetry might be and structure, because far-out planets will generally be much particularly valuable when searching for extrasolar planets with cooler than planets close to their star. Earth-like atmospheres. Our simulations show that starlight that is reflected by Jupiter-like EGPs can have degrees of polarization of several tens of percent, thanks to the range of phase angles α at which References ◦ EGPs will generally be observable from Earth (between 90 − i Burrows, A., Sudarsky, D., & Lunine, J. I. 2003, ApJ, 596, 587 ◦ and 90 + i, with i the orbital inclination angle). Only when Charbonneau, D., Brown, T. M., Latham, D. W., & Mayor, M. 2000, α ≈ 0◦ or 180◦, P will be close or equal to zero. At those phase ApJ, 529, L45 angles, it will be very difficult to spatially resolve the planet Charbonneau, D., Brown, T. M., Noyes, R. W., & Gilliland, R. L. from the star. 2002, ApJ, 568, 377 Because P is a relative measure, it can be determined with De Haan, J. F., Bosma, P. B., & Hovenier, J. W. 1987, A&A, 183, 371 accuracies of the order of 10−5 (Povel 1995). Although such De Rooij, W. A., & van der Stap, C. C. A. H. 1984, A&A, 131, 237 sensitivities are imperative for planet detection in unresolved Feldt, M., Turatto, M., Schmid, H. M., et al. 2003, In ESA SP–539, / planetary systems, the detection of a spatially resolvable EGP Towards Other Earths: DARWIN TPF and the Search for Extrasolar Terrestrial Planets, ed. M. Fridlund, & Th. Henning, in a Jupiter-like orbit can be carried out with a less accu- 99 rate polarimeter. However, the use of an accurate polarimeter Hansen, J. E., & Hovenier, J. W. 1974, J. Atmos. Sci., 31, 1137 is strongly recommended, because polarimetry is a valuable Hansen, J. E., & Travis, L. D. 1974, Space Sci. Rev., 16, 527 tool not only for detecting EGPs, but also for the characteriza- Hough, J. H., & Lucas, P. W. 2003, In ESA SP–539, Towards Other tion of their atmospheres, i.e. the derivation of chemical com- Earths: DARWIN/TPF and the Search for Extrasolar Terrestrial positions, sizes, and spatial distributions of atmospheric con- Planets, ed. M. Fridlund, & Th. Henning, 11 stituents, such as cloud particles. Hovenier, J. W. 1969, J. Atmos. Sci., 26, 488 672 D. M. Stam et al.: Polarimetry to detect and characterize Jupiter-like extrasolar planets

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